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Accurate interpolation and approximation techniques for functions with discontinuities are key tools in many applications as, for instance, medical imaging. In this paper, we study an RBF type method for scattered data interpolation that incorporates discontinuities via a variable scaling function. For the construction of the discontinuous basis of kernel functions, information on the edges of the interpolated function is necessary. We characterize the native space spanned by these kernel functions and study error bounds in terms of the fill distance of the node set. To extract the location of the discontinuities, we use a segmentation method based on a classification algorithm from machine learning. The conducted numerical experiments confirm the theoretically derived convergence rates in case that the discontinuities are a priori known. Further, an application to interpolation in magnetic particle imaging shows that the presented method is very promising.

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In this paper we address the problem of approximating functions with discontinuities via kernel-based methods. The main result is the construction of discontinuous kernelbased basis functions. The linear spaces spanned by these discontinuous kernels lead to a very flexible tool which sensibly or completely reduces the well-known Gibbs phenomenon in reconstructing functions with jumps. For the new basis we provide error bounds and numerical results that support our claims. The method is also effectively tested for approximating satellite images.

In this work, we extend the so-called mapped bases or fake nodes approach to the barycentric rational interpolation of Floater-Hormann and to AAA approximants. More precisely, we focus on the reconstruction of discontinuous functions by the S-Gibbs algorithm introduced in De Marchi et al. (2020). Numerical tests show that it yields an accurate approximation of discontinuous functions.

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Aims. In this study, we introduce a general paradigm for generating independent and well-balanced training, validation, and test sets for use in supervised machine and deep learning flare forecasting, to determine the extent to which video-based deep learning can predict solar flares. Methods. We use this implementation paradigm in the case of a deep neural network, which takes videos of magnetograms recorded by the Helioseismic and Magnetic Imager onboard the Solar Dynamics Observatory (SDO/HMI) as input. Results. The way the training and validation sets are prepared for network optimization has a significant impact on the prediction performances. Furthermore, deep learning is able to realize flare video classification with prediction performances that are in line with those obtained by machine learning approaches that require an a priori extraction of features from the HMI magnetograms. Conclusions. To our knowledge, this is the first time that the solar flare forecasting problem is addressed by means of a deep neural network for video classification, which does not require any a priori extraction of features from the HMI magnetograms.

A concept of total stability for continuous or discrete dynamical systems and a generalized definition of bifurcation are given: it is possible to show the link between an abrupt change of the asymptotic behaviour of a family of flows and the arising of new invariant sets, with determined asymptotic properties. The theoretical results are a contribution to the study of the behaviour of flows near an invariant compact set. They are obtained by means of an extension of Liapunov's direct method. © 1975 Fondazione Annali di Matematica Pura ed Applicata